Trigonometric Functions of Angles Unit 5 Trigonometric Functions of Angles

7.1 Angle Measure

What is an angle? Angle: 2 rays with a common vertex How many angles do I have here? Positive Angle  Negative Angle 

More about angles: An angle is in “standard” position when its initial side is the positive x-axis and its vertex is at the origin. Positive angles are formed in a counter-clockwise direction and negative angles are formed in clockwise direction.

What is an angle measured in? Radians One radian is the measure of the central angle (Theta-Θ) whose arc (s) is equal to the length of the radius (r)

What is the circumference of a circle with a radius of 1 unit? 2 So 1 revolution around the coordinate plane = ? Therefore 1 revolution around the coordinate plane = 2 radians

Radians -3/2 /2  2 - 3/2 -/2

Sketch the angle (find coterminal angles) Coterminal angles are angles that have the same terminal side: 2/3 -/4 7/5 7/3

What is an angle measured in – Part II Degrees What is the difference between 3 and 3? 1 = 1/360 of a circle Why?

Sketch the angle and find coterminal angles: Degrees 150 282 -60 -150 450

Conversion Degrees to Radians Multiply by /180 Radians to Degrees

Convert the following 150 282 -60 -150 450 2/3 -/4 7/5 7/3

Arc Length – s = r  where s is the arc length, r is the radius and  is the angle in Radians If r = 4 in find arc length if  = 240 If r = 8 in and s = 15 in, find the angle

Area of a Sector What do you think is true about Θ here? Find the area of a sector with central angle 60∘ if the radius of the circle is 3m. The area of a sector of a circle with a central angle of 2 rad is 16 m2. What is the radius of the circle?

Homework 7.1 Pg 453-454: 2 – 50 even, 50 – 62 even, 45

Right Triangle Trigonometry

Right Triangle Trigonometry hyp opp  adj

Right Triangle Trig: SOH CAH TOA

Right Triangle Trigonometry Evaluate all 6 trig functions for . 3  4

Sketch the triangle: cot  = 5 cos = 3/7

Special Right Triangle: /3 /4 /6 /4

Solving Triangles – solve for unknowns 15 50

Solving Triangles – solve for unknowns 32 10

Solving Triangles – solve for unknowns 25 12

Homework 7.2a Pg 462 – 463: 1 – 4, 11 – 16, 22 – 25, 27, 28

Right Triangle Trigonometry – story Problems

A kite is held at a 75 angle using 300 ft of string, how high off the ground is the kite if the person holds the kite 5 ft off the ground?

300 x 75 5 ft

A ladder is placed so that it reaches a point 8 feet from the ground on a wall, if the ladder makes a 15 with the wall, how long is the ladder?

15 x 8

A 40 foot ladder leans against a building A 40 foot ladder leans against a building. If the base of the ladder is 6 ft from the base of the building, what is the angle formed by the ladder and the building?

Farmer John looks down into a valley and sees his favorite cow Eugene grazing on grass. He notices that a balloon is floating directly above Eugene (his favorite cow). Farmer John determines that the angle of elevation to the balloon is 72 and the angle of depression to Eugene is 30, find how far above Eugene the balloon is if the hill is 40 ft from the valley below (and Farmer John’s eyes are 5 feet off the ground)

y x 72 Horizontal line 30 45 ?

From a point on the ground 500 ft from the base of a building, it is observed that the angle of elevation to the top of the building is 24 degrees, and the angle of elevation to the top of the flagpole atop the building is 27 degrees. Find the height of the building and the length of the flagpole.

Homework 7.2b Pg 463 464: 34 – 46 even

Law of Sines How do I solve non-right triangles? 7.4 Law of Sines How do I solve non-right triangles?

Laws of Sines – angle and an opposite side Given B a c C A b

Solve: B a c 25  20  C A 80.4

Solve: B a 32 100  50  C A b

But wait – there is a problem when you know 2 sides and an opposite angle What is the sin30? ½ What is the sin150? When you do sin-1(1/2) how do you know if its supposed to be 30 or 150?

Scenario I: B 7 c 45  C A 9.9

Scenario II: B 248 186 43  C A b

Scenario III: B 122 42  32 C A 70

7.4a Homework Pg 483-484: 1, 4, 6, 10, 16, 21, 22

One, Two or Zero?:

Law of Sines – story problems

Story Problem #1 The course for a boat race starts at a point A and proceeds in the direction of S52W to point B, then in a direction of S40E to point C, and finally back to A (due North). If A and C are 8 km apart, what is the total distance for the race?

Story Problem #2 The pitchers mound on a softball field is 46 feet from home plate and the distance from the pitchers mound to first base is 42.6 feet. How far is first base from home plate?

Story Problem #3 A hot air balloon is flying above High Point. To the left side of the balloon, the balloonist measures the angle of depression to a soccer field to be 20⁰ and to the right he sees a football field and finds the angle of depression to be 62.5⁰. If the distance between the two fields is know to be 4 miles, what is the direct distance from the balloon to the soccer field? How high is the balloon?

Story Problem #4 An architect is designing an overhang above a sliding glass door. During the heat of the summer, the architect wants the overhang to prevent the rays of the sun from striking the glass at noon. The overhang is to have an angle of depression equal to 55⁰ and starts 13 feet above the ground. If the angle of elevation of the sun during this time is 63⁰, how long should the architect make the overhang?

Homework 7.4b Pg 484 – 485: 24 – 30 even

7.5 Law of Cosines

Laws of Cosines – angle and the two included sides Given B a c C A b

Solve: B a 212 82  C A 388 Hint always find the smaller angle first when you use the law of sines after using the law of cosines – why?

Solve: B 18 c 47  C A 105

Solve: B 5 8 C A 12 Why should we solve for the biggest angle first? Which is the biggest angle?

Story Problem #1 A ship travels 60 miles due east, then adjust its course 15 northward. After traveling 80 miles how far is the ship from where it departed?

Story Problem #2 A car travels along a straight road, heading east for 1 hour, then traveling for 30 minutes on another road that leads northeast. If the car has maintained a constant speed of 40 mph in what direction did the car turn?

Homework 7.5 Pg 491-492: 1-3, 17-21, 24, 25, 27, 32

Navigation Problem/Worksheet A boat “A” sights a lighthouse “B” in the direction N65⁰E and the spire of a church “C” in the direction S75⁰E. According to a map, the church and lighthouse are 7 miles apart in a direction of N30⁰W. Find the bearing the boat should continue at in order to pass the lighthouse at a safe distance of 4 miles.

Navigation Worksheet

Unit 5 Test Review